Information

The absurd capriciousness underlying such a memory system is best represented by the categorization scheme of an ancient Chinese encyclopedia entitled Celestial Emporium of Benevolent Knowledge, as interpreted by the South American fiction master J. L. Borges.
On those remote pages it is written that animals are divided into (a) those that belong to the Emperor, (b) embalmed ones, (c) those that are trained , (d) suckling pigs, (e) mermaids, (f) fabulous ones, (g) stray dogs, (h) those that are included in this classification, (i) those that tremble as if they were mad, (j) innumerable ones, (k) those drawn with a very fine camel's hair brush, (1) others, (m) those that have just broken a flower vase, (n) those that resemble flies from a distance.

Abstract
We discuss mathematical and physical arguments against continuity and in favor of discreteness, with particular emphasis on the ideas of Emile Borel (1871–1956).

Introduction
Experimental physicists know how difficult accurate measurements are. No physical quantity has ever been measured with more than 15 digits or so of accuracy. Mathematicians, however, freely fantasize with infinite-precision real numbers. Nevertheless within pure math the notion of a real number is extremely problematic.

We’ll compare and contrast two parallel historical episodes:
1. the diagonal and probabilistic proofs that reals are uncountable, and
2. the diagonal and probabilistic proofs that there are uncomputable reals.

Both case histories open chasms beneath the feet of mathematicians. In the first case these are the famous Jules Richard paradox (1905), Emile Borel’s know-it-all real (1927), and the fact that most reals are unnameable, which was the subject of Borel, 1952, his last book, published when Borel was 81 years old James, 2002. In the second case the frightening features are the unsolvability of the halting problem (Turing, 1936), the fact that most reals are uncomputable, and last but not least, the halting probability , which is irreducibly complex (algorithmically random), maximally unknowable, and dramatically illustrates the limits of reason Chaitin, 2005.

In addition to this mathematical soul-searching regarding real numbers, some physicists are beginning to suspect that the physical universe is actually discrete Smolin, 2000 and perhaps even a giant computer Fredkin, 2004, Wolfram, 2002. It will be interesting to see how far this so-called “digital philosophy,” “digital physics” viewpoint can be taken.

General systems theorists often refer to matter, energy and information as fundamental categories. The three concepts – matter, energy and information – are related through scientific laws. Matter and energy relations are more thoroughly understood than relations involving information. At the level of data or signal “difference” is suggested as a more elementary term than “information.”

Stuart A. Umpleby
Department of Management Science
The George Washington University
Washington, DC 20052 USA

Information is the difference that makes a difference.
- Gregory Bateson